ÖMO 2016 Problem 6

Let $a,b,c$ be three integers for which the sum $$\frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}$$ is integer. Prove that each of the three numbers $$\frac{ab}{c}, \frac{ac}{b}, \frac{bc}{a}$$ is integer.

(Proposed by Gerhard J. Woeginger)